Coming back to the case \(\alpha <0\) , a first regularity result is proved in [1]} for solutions of the homogeneous Dirichlet problem. When the operator \(F\)  is concave or convex, the \({\cal C}^2\)  regularity holds.
In the case \(\alpha >0\)  and \(h = 0\) , Imbert and Silvestre proved a \({\cal C}^{1 , \beta }\)  interior regularity result in [2]}. This result is extended to a local result "up to the boundary" and to the case where \(h\ne 0\) , [3]}.
The interior regularity is precised in [4]} in the case \(\alpha >0\) , which can be \({ \cal C}^{1,{1\over 1+ \alpha }}\)  in the cases where for example \(F\)  is convex or concave .
